For the continuous time case, many diffusion models involving tempered fractional calculus were established and the solvability of the resulting equation becomes important and difficult. To achieve this aim, a lot of work has been carried out. A Chebyshev pseudospectral scheme was developed to discretize the space-time tempered fractional diffusion equation in [1]}. A time discretization method was established for approximating the mild solution of the tempered fractional Feynman–Kac equation in [2]}. An efficient and stable finite difference scheme was proposed for solving space tempered fractional diffusion equations in [3]}. The tempered Lévy flights were introduced to process the anomalous diffusion problem in [4]}. The equivalence between the tempered fractional derivative operator and the Hadamard finite-part integral was first proved and then the fractional linear multistep method was extended to the tempered fractional integral and derivative operators in [5]}. A class of tempered fractional neural networks was proposed and the conditions for attractivity and Mittag–Leffler stability were provided in [6]}.
